Let $I$ be an ideal of the polynomial ring $P=K[x_{1},...,x_{n}]$ that is generated by degree two polynomials ${f_1,...,f_k}$. The zero set $\mathcal{Z}(I)$ is isomorphic to an affine space of dimension $m,$ where $m<n$.

Let $Y=\{y_1,...,y_m\}\subset \{x_1,...,x_n\}$ and consider the isomorphism below,

$$P/I \rightarrow K[Y].$$

Is it straightforward to say that $I$ can be generated by $n-m$ polynomials?

To rephrase my question, I have a smooth, connected and irreducible variety. Is it straightforward to say it is an ideal theoretic complete intersection?

I am sorry if i am sloppy in my description.

Let $I$ be an ideal of the polynomial ring $P=K[x_{1},...,x_{n}]$ that is generated by degree two polynomials ${f_1,...,f_k}$. The zero set $\mathcal{Z}(I)$ is isomorphic to an affine space of dimension $m,$ where $m<n$. Let $\mathfrak{m}$ be the maximal ideal generated by $\{\bar{x}_1,...,\bar{x}_n\}$ in $P/I.$

Let $Y=\{y_1,...,y_m\}\subset \{x_1,...,x_n\}$ where the equivalence classes of $y_1,...,y_m$ form a basis of $\mathfrak{m}/\mathfrak{m}^2.$ Consider the isomorphism below,

$$P/I \rightarrow K[Y].$$

Is it straightforward to say that $I$ can be generated by $n-m$ polynomials?

To rephrase my question, I have a smooth, connected and irreducible variety. Is it straightforward to say it is an ideal theoretic complete intersection?

I am sorry if i am sloppy in my description.